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Thermoelectric cooling uses the Peltier effect to remove heat from a load using electrical work. A typical thermoelectric module connects many pairs of different semiconductors electrically in a series circuit and thermally in parallel between two faces. For an idealized pair with $S_p>S_n$, positive conventional current crosses n to p at the cold bridge and p to n at the hot return. With common bridge material and isothermal contacts, the intervening metal contributions telescope: the net pair coefficient is $S_p-S_n$. For $N$ identical pairs define the module Seebeck coefficient $\alpha =N(S_p-S_n)>0$.

Choose fixed face temperatures $T_h\ge T_c>0$, $\Delta T=T_h-T_c$, module electrical resistance $R>0$ and open-circuit thermal conductance $K>0$. All three module parameters $\alpha ,R,K$ are constants over the chosen operating range. Positive $I$ is the orientation absorbing Peltier effect heat at c. Define $Q_c$ as net heat extracted from the cold reservoir per unit time, $Q_h$ as net heat delivered to the hot reservoir per unit time, and $P_{\mathrm{in}}$ as electrical power entering the module. Despite the conventional letters $Q_c,Q_h$, both denote power in watts, not accumulated energy. Ferrotec gives the corresponding single-stage equations and distinguishes module performance from bypass leaks.^[ref]

For a constant-property thermoelectric module consisting of two material legs, let ${\alpha }_{np}=S_p-S_n$ be their Seebeck coefficient difference, $R$ their total series electrical resistance, and $K$ their total parallel thermal conductance. The couple's thermoelectric figure of merit isIts units are ${\mathrm{K}}^{-1}$; multiply by an absolute temperature to form a dimensionless figure. This expression assumes that contact losses are absent or consistently included in $R$ and $K$.

For equal leg lengths $l$ and areas $A$, with equal electrical resistivity $\rho$ and thermal conductivity $\kappa$, one has $R=2\rho l/A$ and $K=2\kappa A/l$. Consequently $Z_{\mathrm{pair}}={\alpha }_{np}^2/(4\rho \kappa )$. The factor four follows from combining two series resistances and two parallel conductances; it disappears only if the symbols have already been defined as pair properties. With opposite equal Seebeck coefficients $S_p=-S_n=S$, the result reduces to $S^2/(\rho \kappa )$.

A thermoelectric module connects thermoelectric material pairs between thermal faces. Electrical series connections and thermal parallel connections combine their transport responses for thermoelectric cooling or electrical generation.